Logic

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This is one of the problems for my exam revision

Prove that Phi is a tautology using logical identities only, without using a truth table

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<@&286206848099549185>

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<@&286206848099549185>

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ϕ = ((p ∨ q) ∧ (¬p ∨ r)) → (q ∨ r)
≡ ¬((p ∨ q) ∧ (¬p ∨ r)) ∨ (q ∨ r)
≡ ¬(p ∨ q) ∨ ¬(¬p ∨ r) ∨ (q ∨ r)
≡ (¬p ∧ ¬q) ∨ (p ∧ ¬r) ∨ (q ∨ r)
≡ (¬p ∧ ¬q) ∨ (p ∧ ¬r) ∨ q ∨ r
≡ q ∨ r ∨ (¬p ∧ ¬q) ∨ (p ∧ ¬r)
≡ [q ∨ (¬p ∧ ¬q)] ∨ [r ∨ (p ∧ ¬r)]
≡ (q ∨ ¬p) ∨ (r ∨ p)
≡ q ∨ ¬p ∨ r ∨ p
≡ (¬p ∨ p) ∨ q ∨ r
≡ True ∨ q ∨ r
≡ True

Thus ϕ is a tautology

This is the right solution

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